Hierarchical ordination
2024-01-03
An Example of Hierarchical Ordination
This document describes fitting a hierarchical ordination to data, including code and the details that are needed.
The data
The data is available on figshare, but was originally collected by Ribera et al. (2001), and reanalyzed by Niku et al. (2021). It includes abundance observations of 68 ground beetles 87 sites in Scotland, along with 17 environmental variables and 20 traits. The aim to is determine how the environmental variables and traits affect composition in the ecological community, including whether they interact. The methodological question is how does hierarchical ordination help.
The abundances, traits and environment are each stored in a different matrix. First we load the data and set some constants:
library(nimble)
library(nimbleHMC)
library(coda)
library(lattice)
# Response data
Y <- t(read.csv("Y.csv"))
colnames(Y) <- Y[2,]
Y<-Y[-c(1:2),-c(1,70:71)]
Y <- as.data.frame(apply(Y,2,as.integer))
# Environmental predictors
X <- read.csv("X.csv")[,-c(1:5)]
X <- as.data.frame(apply(X,2,as.numeric))
X$Sampling.year <- X$Sampling.year - min(X$Sampling.year)
X$Texture <- as.factor(X$Texture)
# Traits
TR <- read.csv("TR.csv")
row.names(TR) <- TR$SPECIES
TR <- TR[,-c(1:3)]
# Traits to categorical
# Removing question marks, not ideal
TR[,c("CLG","CLB","WIN","PRS","OVE","FOA","DAY","BRE","EME","ACT")] <- apply(TR[,c("CLG","CLB","WIN","PRS","OVE","FOA","DAY","BRE","EME","ACT")],2,function(x)as.factor(gsub("\\?.*","",x)))
# Data standardization
X <- scale(model.matrix(~.,X))[,-1] # environmental variables
TR <- scale(model.matrix(~.,TR))[,-1] # species traits
# Constants
NSites <- nrow(Y) # number of sites
NSpecies <- ncol(Y) # number of species
NTraits <- ncol(TR) # number of traits
NEnv <- ncol(X) # number of environmental predictors
# create data lists for Nimble
dat <- list(Y = Y, X = as.matrix(X), TR = as.matrix(TR))
consts <- list(NSites = NSites, NEnv = NEnv, NTraits = NTraits, NSpecies = NSpecies)The Model
The data are counts of each species so we assume they follow a Poisson distribution with a log link function, as we would do in a standard generalised linear model. We assume that each species has a different mean abundance (i.e. for each species \(j\) we have a different intercept \(\beta_{0j}\)), and model the rest of the variation with a hierarchical ordination. This gives the following mean model on the link scale (with linear predictor \(\eta_{ij}\)):
\[ \eta_{ij} = \beta_{0j} + \boldsymbol{z}_i^\top \boldsymbol{\Sigma} \boldsymbol{\gamma}_j. \]
As with any ordination, \(\boldsymbol{z}\) and \(\boldsymbol{\gamma}_j\) are the site scores and species loadings, and the columns of \(\boldsymbol{Z}\) are the latent variables (which holds the site scores as the rows). Here we assume that they each have a variance of one, so that \(\boldsymbol{\Sigma}\) holds the variation of the latent variables: it will typically be a diagonal matrix, and if any of the terms on the diagonal are close to zero, this suggests that latent variable has (almost) no effect. It thus provides a straightforward summary of the relative importance of that latent variable, and is similar to the square root of eigenvalues (singular values) in an eigenanalysis.
If we stopped here, this would be a standard Generalized Linear Latent Variable Model. But, here we want to model both \(\boldsymbol{z}_i\) and \(\boldsymbol{\gamma}_j\), i.e. go from modelling model groups of species responding in similar ways to sites to modelling how the traits of the species affect their responses to the environment.
As a simplification, we can think about this as simply a regression of \(\boldsymbol{z}_i\) (the site effects) against the environmental variables and \(\boldsymbol{\gamma}_j\) against the traits. In reality, it is more complicated, because \(\boldsymbol{z}_i\) and \(\boldsymbol{\gamma}_j\) are estimated in the model, so their uncertainty needs to be propagated.
Some more mathematical details are hidden away here, for those who are interested.
We denote the abundance at site \(i = 1 \dots n\) for species \(j = 1 \dots p\) as \(Y_{ij}\), or as a matrix as \(\boldsymbol{Y}\). The environmental variables are \(x_{ik}\) for the \(k = 1 \ldots K\) predictors (or \(\boldsymbol{X}\) as a matrix), and \(t = 1\ldots T\) traits as \(\boldsymbol{TR}\). We then assume
\[Y_{ij} \sim \text{Pois}(\lambda_{ij}),\]
with
\[\text{log}(\lambda_{ij}) = \eta_{ij}.\]
Consequently, \(\eta_{ij}\) is the linear predictor, which we further model with the hierarchical ordination:
\[ \eta_{ij} = \beta_{0j} + \boldsymbol{z}_i^\top \boldsymbol{\Sigma} \boldsymbol{\gamma}_j. \]
We then model \(\boldsymbol{z}_i\) (as in van der Veen et al. (2023)) and \(\boldsymbol{\gamma}_j\) hierarchically:
\[ \boldsymbol{z}_i = \boldsymbol{B}^\top\boldsymbol{x}_i + \boldsymbol{\epsilon}_i \] and
\[ \boldsymbol{\gamma}_j = \boldsymbol{\omega}^\top\boldsymbol{TR}_{j} + \boldsymbol{\varepsilon}_j \] where:
- \(x_{ik}\) is the \(k^{th}\) predictor (i.e. environmental effect) at site \(i\)
- \(\boldsymbol{B}\) with entry \(b_{kq} \sim \mathcal{N}(0,1)\) is the effect of the \(k^{th}\) predictor on the site score for the \(q^{th} = 1\ldots d\) latent variable
- \(\boldsymbol{\epsilon}_i\) with entry \(\epsilon_{iq} \sim \mathcal{N}(0, \sigma^2_q)\) is a vector of residuals for the unexplained part of the site score
- \(TR_{jt}\) is the \(t^{th}\) predictor (i.e. trait) for species \(j\)
- \(\boldsymbol{\omega}_t\) with entry \(\omega_{tq}\) is the effect of the \(t^{th}\) trait on the species loading for the \(q^{th}\) latent variable
- \(\boldsymbol{\varepsilon}_j\) with entry \(\varepsilon_{jq} \sim \mathcal{N}(0, \delta^2_q)\) is a vector of residuals for the unexplained part of the species loading
Implementation
We fit the model with the Nimble R-package. We start with a single dimension for simplicity, so that we can show the steps needed.
Functions needed for this vignette are here.
Functions for MCMC are here.
# Function to run one chain: it can be done with HMC or other MCMC algorithms.
# "block" can be used to specify blocking structures for the slice sampler
# "slice" can be used to specify parameters on which to apply univariate Metropolis-Hastings
# parameters that are not in "block" or "slice" are HMCed
# Function to run a single chain
RunOneChain <- function(seed, dat, code, inits, consts, ToMonitor=NULL,
Nburn=5e3, NIter=5.5e4, Nthin=10, block = NULL, slice = NULL, ...) {
require(nimble)
require(nimbleHMC)
AllSamplers <- HMCsamplers <- c('epsilonSTAR', 'varepsilonSTAR', 'beta0', 'OSTAR', 'BSTAR',
'sd.SiteSTAR', 'sd.SpeciesSTAR','xi', 'phi')
if(!is.null(block)){
HMCsamplers <- HMCsamplers[!HMCsamplers%in%unique(gsub("\\s*\\[[^\\)]+\\]","",c(unlist(block),unlist(slice))))]
}
if(is.null(ToMonitor)) {
ToMonitor <- c("beta0", "sd.SpeciesSTAR", "sd.SiteSTAR", "sd.LV", "BSTAR", "OSTAR", "B","O",
"epsilonSTAR", "varepsilonSTAR", "xi", "phi")
}
mod <- nimbleModel(code = code, name = "HO", inits = inits(consts), constants = consts, data = dat, buildDerivs = TRUE)
model <- compileNimble(mod)
# Do HMC
conf <- configureHMC(model, nodes = HMCsamplers, monitors = ToMonitor, print = FALSE,
control=list(nwarmup=Nburn))
if(!is.null(block)) {
if(is.list(block)){
# Use a slice everything that not being HMCed
lapply(block, conf$addSampler, type = "AF_slice", sliceAdaptFactorInterval = Nburn)
}else{
# Use a slice everything that not being HMCed
sapply(block, conf$addSampler, type = "AF_slice", sliceAdaptFactorInterval = Nburn)
}
}
if(!is.null(slice)) {
if(is.list(slice)){
# Use a slice everything that not being HMCed
lapply(slice, conf$addSampler, type = "slice")
}else{
# Use a slice everything that not being HMCed
sapply(slice, conf$addSampler, type = "slice")
}
}
mcmc <- buildMCMC(conf)
cmcmc <- compileNimble(mcmc, project = model)
res <- runMCMC(cmcmc, niter=NIter, nburnin = Nburn, thin=Nthin,
nchains = 1, samplesAsCodaMCMC = TRUE)
return(res)
}
# Function to run chains in parallel
ParaNimble <- function(NChains, ...) {
opts <- list(...)
if(!is.null(opts$seeds) && (length(opts$seeds) == NChains)){
seeds <- opts$seeds
opts <- opts[-which(names(opts)=="seeds")]
}else{
seeds <- 1:NChains
}
require(parallel)
nimble_cluster <- makeCluster(NChains)
clusterExport(nimble_cluster, "nimQR.u")
samples <- parLapply(cl = nimble_cluster, X = seeds, ...)
stopCluster(nimble_cluster)
# Name the chains in the list
chains <- setNames(samples,paste0("chain", 1:length(samples)))
chains
}
# Function to create list of names for parameters to use a block slice sampler for
MakeBlockList <- function(consts, LVwise = TRUE){
# builds list for slice AF sampling
# with LVwise = TRUE we are LV-wise blocking B and epsilon, and O and varepsilon
# otherwise, LVs are jointly blocked
if(LVwise & consts$nLVs>1){
blockList <- c(sapply(1:consts$nLVs,
function(x,consts){
c(paste0("BSTAR[", 1:consts$NEnv,", ",x, "]"),
paste0("epsilonSTAR[", 1:consts$NSites,", ",x, "]"))
}
,
consts = consts,simplify=F),
sapply(1:consts$nLVs,
function(x,consts){
c(paste0("OSTAR[", 1:consts$NTraits,", ",x, "]"),
paste0("varepsilonSTAR[", 1:consts$NSpecies,", ",x, "]"))
}
,
consts = consts,simplify=F)
)
}else{
blockList = list(c("BSTAR","epsilonSTAR"),c("OSTAR","varepsilonSTAR"))
}
blockList
}Functions for processing the MCMC are here.
GetMeans <- function(summ, name, d) {
var <- summ$statistics[grep(paste0('^', name), rownames(summ$statistics)),"Mean"]
matrix(var,ncol=d)
}
# Utility Function to get logical indicators for if names contains a string in v
GetInds <- function(v, names) {
if(length(v)==1) {
res <- grep(v, names)
} else {
res <- c(unlist(sapply(v, grep, x=names)))
}
res
}
# Function to swap signs of all variables varinds to have same sign as vartosign.
ReSignChain <- function(chain, varinds, vartosign) {
# MeanSign <- sign(mean(chain[ ,s])) # Might need this
res <- t(apply(chain, 1, function(x, vs, vi) {
Names <- names(x)[vi]
if(any(grepl(",", Names))) {
lvind <- gsub(".*, ", "", Names)
} else {
lvind <- seq_along(vs)
}
x[vi] <- x[vi]*sign(x[vs[lvind]])
x
}, vs=vartosign, vi=varinds))
as.mcmc(res)
}
# Function to post-process chains to swap signs.
postProcess <- function(Chains, VarsToProcess, VarsToSwapBy = NULL, VarToSign=NULL, print=FALSE, rule = 2) {
if(is.null(VarToSign)) VarToSign <- VarsToProcess
SignInd <- GetInds(VarToSign, colnames(Chains[[1]]))
# Get indicators for all variables to have their signs changed
ProcessInds <- GetInds(VarsToProcess, names = colnames(Chains[[1]]))
# Check if > 1 LV
SeveralLVs <- any(grepl(",", colnames(Chains[[1]])[SignInd]))
LV <- gsub(".*, ", "", colnames(Chains[[1]])[SignInd])
if(rule==1){
# Calculate variance of mean of indicator of sign:
# hopefully largest is variable with most sign swapping (i.e. )
Signs <- as.data.frame(lapply(Chains, function(mat, ind) {
colMeans(mat[,ind]>0)
}, ind=SignInd))
VarSign <- apply(Signs, 1, var)
if(SeveralLVs) {
LV <- gsub(".*, ", "", colnames(Chains[[1]])[SignInd])
#Chose variables who's sign will be used to swap other signs
if(is.null(VarsToSwapBy)){
VarsToSwapBy <- sapply(unique(LV), function(lv, vs) {
vv <- vs[grep(lv, names(vs))]
nm <- names(which(vv==max(vv)))
if(length(nm)>1) nm <- nm[1] # probably something more sophisticated is better
nm
}, vs = VarSign, simplify = TRUE)
}
} else { # only 1 LV
if(is.null(VarsToSwapBy)){
#Chose variables who's sign will be used to swap other signs
VarsToSwapBy <- names(which(VarSign==max(VarSign)))[1]
}
}
}else if(rule==2){
require(mousetrap)
jointChains <- do.call(rbind, Chains)
# bimodality score
bms<-apply(jointChains[,SignInd],2,bimodality_coefficient)
# find maximum bimodality score
if(SeveralLVs){
lstSgn <- sapply(unique(LV),function(lv)which.max(bms[grep(lv,names(bms))]),simplify=F)
# formatting
names(lstSgn) <- NULL
VarsToSwapBy <- names(unlist(lstSgn))
names(VarsToSwapBy) <- paste0(sort(unique(LV)))
}else{
lstSgn <- which.max(bms)
# formatting
VarsToSwapBy <- names(lstSgn)
names(VarsToSwapBy) <- "1]"
}
}
if(print) message(paste0("Swapping by ", paste(VarsToSwapBy, collapse=", ")))
chains.sgn <- lapply(Chains, ReSignChain, vartosign=VarsToSwapBy, varinds=ProcessInds)
as.mcmc.list(chains.sgn)
}
# Function to convert a variable in an MCMC chain that should be a matrix from
# a vector to the right matrix
ChainToMatrix <- function(ch, name) {
v <- ch[grep(paste0("^", name, "\\["), names(ch))]
l.row <- as.numeric(gsub(".*\\[", "", gsub(",.*", "", names(v))))
l.col <- as.numeric(gsub(".*, ", "", gsub("\\]", "", names(v))))
mat <- matrix(0, nrow=max(l.row), ncol=max(l.col))
mat[l.row+max(l.row)*(l.col-1)]<-v
mat
}
# Function to convert a variable in an MCMC chain that should be a vector to a diagonal matrix
ChainToDiag <- function(ch, name) {
v <- ch[grep(paste0("^", name, "\\["), names(ch))]
mat <- diag(v)
mat
}
RescaleVars <- function(vec, ScaleBy, SDTorescale,ToRescale) {
if(!any(c("z","gamma")%in%ScaleBy))stop("Not a valid choice.")
if("sd.LV"%in%SDTorescale)stop("Not a valid choice.")
vec2 <- vec
# get scale from z or gamma
ScBy <- ChainToMatrix(vec, ScaleBy)
SD <- apply(ScBy, 2, sd)
for(nm in unique(c(ScaleBy, ToRescale))) {
Sc.x <- ChainToMatrix(vec, nm)
Sc.x.sc <- sweep(Sc.x, 2, SD, "/")
vec2[grep(paste0("^", nm, "\\["), names(vec2))] <- c(Sc.x.sc)
}
# scale sd separately
for(nm in SDTorescale) {
SD.x <- vec[grep(paste0("^", nm, "\\["), names(vec))]
vec2[grep(paste0("^", nm, "\\["), names(vec2))] <- SD.x/SD
}
# scale into sd.LV
# sd.LV <- vec[grep("sd.LV", names(vec))]
# vec2[grep("sd.LV", names(vec2))] <- SD*sd.LV
vec2
}
RescaleChains <- function(mcmc.lst, ...) {
rescale <- lapply(mcmc.lst, function(mcmc) {
rot <- apply(mcmc, 1, RescaleVars, ...)
as.mcmc(t(rot))
})
as.mcmc.list(rescale)
}We create a new function to simulate starting values from the prior distributions, which we can hide away
inits <- function(consts){
B = matrix(rnorm(consts$nLVs*consts$NEnv),ncol=consts$nLVs)
O = matrix(rnorm(consts$nLVs*consts$NTraits),nrow=consts$NTraits)
varepsilon = mvtnorm::rmvnorm(consts$NSpecies,rep(0,consts$nLVs),diag(consts$nLVs))
epsilon = mvtnorm::rmvnorm(consts$NSites,rep(0,consts$nLVs),diag(consts$nLVs))
xi = rgamma(1,5,2)
#for(l in 2:consts$nLVs)xi<-c(xi,rbeta(1,consts$NSpecies/(l+consts$NSpecies),l))
xi <- c(xi,rbeta(consts$nLVs-1,1,10))
list(
BSTAR = B,
OSTAR = O,
epsilonSTAR = epsilon,
varepsilonSTAR = varepsilon,
sd.SiteSTAR = rexp(consts$nLVs),
sd.SpeciesSTAR = rexp(consts$nLVs),
beta0 = rnorm(consts$NSpecies),
# sd.LV = rexp(consts$nLVs),
xi = xi,
phi = rep(1, consts$NSpecies)
)
}We use plotting functions, which are hidden here
PlotPost <- function(var, summ, varnames=NULL, ...) {
var <- paste0("^", var, "*\\[")
vars <- grep(var, rownames(summ$statistics))
if(is.null(varnames)) varnames <- rownames(summ$statistics)[grep(var, rownames(summ$statistics))]
if(length(varnames)!=length(vars))
stop(paste0("Number of variable names, ", length(varnames), "not the same as number of variables, ",
length(vars)))
plot(summ$statistics[vars,"Mean"], 1:length(vars),
xlim=range(summ$quantiles[vars,]), yaxt="n", ...)
segments(summ$quantiles[vars,"2.5%"], 1:length(vars), summ$quantiles[vars,"97.5%"], 1:length(vars))
segments(summ$quantiles[vars,"25%"], 1:length(vars), summ$quantiles[vars,"75%"], 1:length(vars), lwd=3)
abline(v=0, lty=3)
axis(2, at=1:length(vars), labels=varnames, las=1)
}
AddArrows <- function(coords, marg= par("usr"), col=2) {
origin <- c(mean(marg[1:2]), mean(marg[3:4]))
Xlength <- sum(abs(marg[1:2]))/2
Ylength <- sum(abs(marg[3:4]))/2
ends <- coords / max(abs(coords)) * min(Xlength, Ylength) * .8
arrows(
x0 = origin[1],
y0 = origin[2],
x1 = ends[,
1] + origin[1],
y1 = ends[, 2] + origin[2],
col = col,
length = 0.1)
text(
x = origin[1] + ends[, 1] * 1.1,
y = origin[2] + ends[, 2] * 1.1,
labels = rownames(coords),
col = col)
}Latent variable models are notorious for being unidentifiable, you can get the same mean abundances from different combinations of the parameters. We have to make some adjustments to the model to account for this: some of this is done in the model fitting, but for others it is easier to do it after we obtain the posterior samples.
The details of what we do to make the HO identifiable are here
- First, we standardise \(\boldsymbol{z}_i\) and \(\boldsymbol{\gamma}_j\) to unit variance per latent variable to prevent scale invariance
- At this point the model is still invariant to sign switching: because \(z_{i}\gamma_{j} = (-z_{i})(-\gamma_{j})\) in the MCMC algorithm can (and does) switch signs mid run. We could solve it by placing a truncated normal prior on the main diagonal entries of \(\boldsymbol{B}\) or \(\boldsymbol{\omega}\), but that results in bimodal posterior distributions for other coefficients. Instead we impose sign constraints by post-process the chains. We choose one parameter, for example one \(z_{i}\) or \(\gamma_{j}\), to be positive, and switch all of the other parameters based on that one. See below for the details of how we fix the signs.
We want to make one parameter positive, so ideally we want to do this to a parameter that has both modes away from zero. We can identify this in an ad hoc way: for each chain for every parameter we calculate the proportion of iterations where the sign is positive, and then for every parameter we calculate the variance in that proportion. If a parameter is centered around 0 the mean proportion will be about 0.5, and will not vary much, whereas if it is some way from 0 the mean will be close to 0 or 1, and the variance will be high. ALternativley, we coiuld look at the largest \(|p - 0.5|\), or we can vissually identify such parameters from the posterior samples. There may be even better alternatives.
Maximum informed dimensions
Since we have two or more latent variables, we also have to worry about their rotation. In the previous example we fixed the rotation by setting some parameters to zero, but here we do this differently.For those who are interested, this is explained further here.
Previously, we set some parameters to zero in the environment effects \(\symbf{B}\) and trait effects \(\symbf{\omega}\) to fix the rotation. Fixing parameters has as advantage that the model is clearly (mathematically) identifiable. However, it has the downside of introducing order dependence in the predictors, traits, and species effects. Bhattacharya and Dunson (2011) developed a multiplicative gamma shrinkage prior which prevents this order dependence, though it leads less clearly to a mathematical identifiability so that (for ordination) the results need to be post-processed so that all MCMC iterations have the same rotation. In this vignette we use a similar prior as Bhattacharya and Dunson (2011), though based on a product of a Gamma and of beta distributions. Use of a beta priors introduces an order constraint to the variance of the latent variables, so that it can only decrease, and has the intuitive interpretation of proportional decrease in variance of one latent variable to the next.
The infinite factor model is motivated by the idea that, although the latent variables may be affected by the rotation, the product of species loadings (i.e., the species associations) is not. The main effects for the predictors, as well as the fourth corner term, are invariant to the rotation and can be estimated even when the model is unidentifiable. The choice of the number of latent variables may be more important in order for an accurate representation of patterns.
First we introduce the additional \(d-\)sized vector \(\symbf{\xi}\), with \(\xi_1 \sim \Gamma(5, 2)\) and \(\xi_{q \in d \setminus \{1\}} \sim B(p/(2q+p),q)\) (or alternatively, to introduce less shrinkage e.g., \(B(5, 5)\) or even \(U(0,1)\)). We use this vector to construct the LV variation parameters as: \(\Sigma_{q,q} = \prod^d_{q=1} \delta_{1:q}\), so that each latent variable explains less variation than the next, as long as the \(\Sigma_{q,q}\) are distinctly different. The choice of gamma distribution is purposefully uninformative, since we do not know how much variation the latent variables explain in the data (but usually, this ranges between 1-5). The choice of parameterization for the beta distribution is motivated as a shrinkage prior; with a low mean and large variance so that the variation for each consecutive latent variable is forced to be considerably lower than that of the previous latent variable. Additionally, by using the index of the latent variable, and the total number of species, as parameters for the prior we ensure that each latent variable receives more shrinkage than the next, and emphasis is placed on a relatively small number of latent variables.We could still estimate some but not all latent variables, for example in larger examples and to speed up computation. The maximum number of latent variables that can be informed (see van der Veen et al. (2023)) by the predictor variables is determined by the number of traits and environmental predictors that we have. In essence, we cannot have more latent variables that the minimum number of traits and environmental predictors, because at that point we have reached the maximum amount of information that (one of, or the combination of) the matrices can explain in the responses (i.e., one of the matrices alone could explain more variation still). In the example here, there are five environmental predictors, but seven traits. Thus, the maximum number of latent variables that can be informed by the matrices jointly is five, though two more dimensions could be informed by the traits alone.
The model code is included here
# Model code
# Update our constants from before with the new number of LVs, rest remains the same
HO.nim <- nimbleCode({
for (i in 1:NSites) {
for (j in 1:NSpecies) {
Y[i, j] ~ dnegbin(phi[j]^-1/(phi[j]^-1+lambda[i, j]), phi[j]^-1)
}
}
ones[1:NSites] <- rep(1, NSites)
# linear predictor
log(lambda[1:NSites, 1:NSpecies]) <- eta[1:NSites, 1:NSpecies]
eta[1:NSites,1:NSpecies] <- asCol(ones[1:NSites])%*%asRow(beta0[1:NSpecies]) +z[1:NSites,1:nLVs]%*%Sigma[1:nLVs,1:nLVs]%*%t(gamma[1:NSpecies,1:nLVs])
Sigma[1:nLVs, 1:nLVs] <- diag(sd.LV[1:nLVs])
# sites
XB[1:NSites,1:nLVs] <- X[1:NSites,1:NEnv]%*%B.ort[1:NEnv,1:nLVs]
zSTAR[1:NSites,1:nLVs] <- XB[1:NSites,1:nLVs] + epsilonSTAR[1:NSites,1:nLVs]%*%diag(sd.SiteSTAR[1:nLVs])
# species
gammaSTAR[1:NSpecies,1:nLVs] <- omegaTR[1:NSpecies,1:nLVs] + varepsilonSTAR[1:NSpecies,1:nLVs]%*%diag(sd.SpeciesSTAR[1:nLVs])
omegaTR[1:NSpecies,1:nLVs] <- TR[1:NSpecies,1:NTraits]%*%O.ort[1:NTraits,1:nLVs]
# prior for intercept
beta0[1:NSpecies] ~ dmnorm(zeroNSpecies[1:NSpecies], prec = diagNSpecies[1:NSpecies,1:NSpecies])
# prior for LV scales
xi[1] ~ dgamma(5,2)
for(l in 2:nLVs) {
xi[l] ~ dbeta(1, 10)#NSpecies/(2*l+NSpecies), l^2) #dbeta(1,20)# could instead be dunif(0,1) to be less informative
}
# diagonal matrices and such for use in priors
diagNTraits[1:NTraits,1:NTraits] <- diag(NTraits)
diagNEnv[1:NEnv,1:NEnv] <- diag(NEnv)
zeroNTraits[1:NTraits] <- rep(0, NTraits)
zeroNEnv[1:NEnv] <- rep(0, NEnv)
diagNSites[1:NSites,1:NSites] <- diag(NSites)
diagNSpecies[1:NSpecies,1:NSpecies] <- diag(NSpecies)
zeroNSites[1:NSites] <- rep(0, NSites)
zeroNSpecies[1:NSpecies] <- rep(0, NSpecies)
for(l in 1:nLVs) {
# prors for LV-level errors
epsilonSTAR[1:NSites,l] ~ dmnorm(zeroNSites[1:NSites], prec = diagNSites[1:NSites,1:NSites])# Residual
varepsilonSTAR[1:NSpecies,l] ~ dmnorm(zeroNSpecies[1:NSpecies], prec = diagNSpecies[1:NSpecies,1:NSpecies]) # Residual
# priors for predictor effects
OSTAR[1:NTraits,l] ~ dmnorm(zeroNTraits[1:NTraits], prec = diagNTraits[1:NTraits,1:NTraits])
BSTAR[1:NEnv,l] ~ dmnorm(zeroNEnv[1:NEnv], prec = diagNEnv[1:NEnv,1:NEnv])
# priors for scale parameters
sd.SiteSTAR[l] ~ dexp(1)
sd.SpeciesSTAR[l] ~ dexp(1)
# constructing LV scale
sd.LV[l] <- prod(xi[1:l])
## standardizing z and gamma for identifiability
## and B and O to have non-unit norms
## rescale the rest of the parameters for trace plots and such
## these are the quantities that really go into eta
#norm.B[l] <- sqrt(sum(BSTAR[1:NEnv,l]^2))
#norm.O[l] <- sqrt(sum(OSTAR[1:NTraits,l]^2))
StdDev.z[l] <- sd(zSTAR[1:NSites,l])
z[1:NSites,l] <- zSTAR[1:NSites,l]/StdDev.z[l]
epsilon[1:NSites,l] <- epsilonSTAR[1:NSites,l]/StdDev.z[l]
B[1:NEnv,l] <- B.ort[1:NEnv,l]/StdDev.z[l]#*norm.B[l]
sd.Site[l] <- sd.SiteSTAR[l]/StdDev.z[l]
StdDev.gamma[l] <- sd(gammaSTAR[1:NSpecies,l])
gamma[1:NSpecies,l] <- gammaSTAR[1:NSpecies,l]/StdDev.gamma[l]
varepsilon[1:NSpecies,l] <- varepsilonSTAR[1:NSpecies,l]/StdDev.gamma[l]
O[1:NTraits,l] <- O.ort[1:NTraits,l]/StdDev.gamma[l]#*norm.O[l]
sd.Species[l] <- sd.SpeciesSTAR[l]/StdDev.gamma[l]
}
# Orthogonalize B and O for identifiability
# first predictor effects B by its F norm
# B.nn[1:NEnv,1:nLVs] <- BSTAR[1:NEnv,1:nLVs]%*%diag(1/norm.B[1:nLVs])
# O.nn[1:NTraits,1:nLVs] <- OSTAR[1:NTraits,1:nLVs]%*%diag(1/norm.O[1:nLVs])
# # get Cholesky factor
# L.B[1:nLVs,1:nLVs] <- chol(t(B.nn[1:NEnv,1:nLVs])%*%B.nn[1:NEnv,1:nLVs])
# L.O[1:nLVs,1:nLVs] <- chol(t(O.nn[1:NTraits,1:nLVs])%*%O.nn[1:NTraits,1:nLVs])
# # Orthogonalize predictor effects
# B.ort[1:NEnv,1:nLVs] <- t(forwardsolve(t(L.B[1:nLVs,1:nLVs]),t(B.nn[1:NEnv,1:nLVs])))
# O.ort[1:NTraits,1:nLVs] <- t(forwardsolve(t(L.O[1:nLVs,1:nLVs]),t(O.nn[1:NTraits,1:nLVs])))
B.ort[1:NEnv,1:nLVs] <- nimQR.u(BSTAR[1:NEnv,1:nLVs], nLVs)
O.ort[1:NTraits,1:nLVs] <- nimQR.u(OSTAR[1:NTraits,1:nLVs],nLVs)
for (j in 1:NSpecies) {
phi[j] ~ dexp(1)
}
})
# QR schwarz-Rutishauser
# The algorithm is correct, but I cannot use it with HMC..
nimQR.u <- nimbleFunction(
run = function(A = double(2), p = double()) {
for (k in 2:p) {
A[, k] <- A[, k]
for (i in 1:(k-1)) {
A[, k] <- A[, k] - inprod(A[,i], A[,k]) / inprod(A[, i], A[, i]) * A[, i]
}
}
return(A)
returnType(double(2))
} ,
buildDerivs = list(run = list(ignore = c("k", "i","p")))
)And now we can run the model:
consts$nLVs <- nLVs <- 10
HO.LV <- ParaNimble(4, fun = RunOneChain,
dat = dat,
code = HO.nim,
inits = inits,
# Nburn=5e1, NIter=5.5e2, Nthin=1, # for a small run
Nburn = 1500, NIter = 15e3, Nthin = 1, # for a full run
consts = consts, block = MakeBlockList(consts, LVwise = FALSE), slice = c(paste0("beta0[",1:consts$NSpecies,"]"),paste0("phi[",1:consts$NSpecies,"]"))) # HMC on the rest (hypers)VarsToProcess <- c("^BSTAR", "^OSTAR", "B", "O", "^epsilonSTAR", "^varepsilonSTAR")
# post-process chains for sign-swapping
chains2LV <- postProcess(HO.LV, VarsToProcess = VarsToProcess, rule = 2, print = TRUE)## Swapping by BSTAR[15, 1], varepsilonSTAR[14, 2], varepsilonSTAR[44, 3], epsilonSTAR[23, 4], varepsilonSTAR[63, 5], B[21, 6], B[24, 7], O[27, 8], B[5, 9], B[5, 10]# Calculate summaries
summ.HOLV <- summary(chains2LV)Results
First, we look at the LV variation. After all, that is what is different in this vignette.
library(basicMCMCplots)
chainsPlot(chains2LV, var = c("sd.LV"), legend = F, traceplot=TRUE)
We can also plot these as a sort of “screeplot”, to see how the variation of the LVs decreases with the number of dimensions:
par(mfrow=c(1,2))
plot(NA,ylim=c(0,max(rbind(summ.HOLV$statistics[grep("sd.LV", row.names(summ.HOLV$statistics)),1],summ.HOLV$quantiles[grep("sd.LV", row.names(summ.HOLV$statistics)),1],summ.HOLV$statistics[grep("sd.LV", row.names(summ.HOLV$statistics)),4]))+0.1),xlim=c(1,consts$nLVs),xlab="Latent variable",ylab="SD")
lines(summ.HOLV$statistics[grep("sd.LV", row.names(summ.HOLV$statistics)),1],type="b",xlab="Latent variable",ylab="SD")
lines(summ.HOLV$quantiles[grep("sd.LV", row.names(summ.HOLV$statistics)),1],type="b",xlab="Latent variable",ylab="SD",col="red",lty="dashed")
lines(summ.HOLV$quantiles[grep("sd.LV", row.names(summ.HOLV$statistics)),4],type="b",xlab="Latent variable",ylab="SD",col="red",lty="dashed")
plot(NA,ylim=c(0,max(rbind(summ.HOLV$statistics[grep("xi", row.names(summ.HOLV$statistics)),1],summ.HOLV$quantiles[grep("xi", row.names(summ.HOLV$statistics)),1],summ.HOLV$statistics[grep("xi", row.names(summ.HOLV$statistics)),4]))+0.1),xlim=c(1,consts$nLVs),xlab="Latent variable",ylab="xi")
lines(summ.HOLV$statistics[grep("xi", row.names(summ.HOLV$statistics)),1],type="b",xlab="Latent variable",ylab="xi")
lines(summ.HOLV$quantiles[grep("xi", row.names(summ.HOLV$statistics)),1],type="b",xlab="Latent variable",ylab="xi",col="red",lty="dashed")
lines(summ.HOLV$quantiles[grep("xi", row.names(summ.HOLV$statistics)),4],type="b",xlab="Latent variable",ylab="xi",col="red",lty="dashed")
Next, we look at the mixing of the predictor coefficients.
Environment first:
library(basicMCMCplots)
chainsPlot(chains2LV, var = c("B"), legend = F, traceplot=TRUE)